Simplify the following expression and state the condition under which the simplification is valid: $p = \dfrac{q^2 + 3q}{q^2 + 8q + 15}$
Solution: First factor the expressions in the numerator and denominator. $ \dfrac{q^2 + 3q}{q^2 + 8q + 15} = \dfrac{(q)(q + 3)}{(q + 5)(q + 3)} $ Notice that the term $(q + 3)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(q + 3)$ gives: $p = \dfrac{q}{q + 5}$ Since we divided by $(q + 3)$, $q \neq -3$. $p = \dfrac{q}{q + 5}; \space q \neq -3$